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<li><a class="reference internal" href="#accessing-elements-within-a-vector">Accessing elements within a vector</a></li>
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  <div class="section" id="introduction-to-vectors-in-matlab">
<span id="vectors"></span><h1>Introduction to Vectors in Matlab<a class="headerlink" href="#introduction-to-vectors-in-matlab" title="Permalink to this headline">¶</a></h1>
<p>This is the basic introduction to Matlab. Creation of vectors is
included with a few basic operations. Topics include the following:</p>
<ol class="arabic simple">
<li><a class="reference internal" href="#definingvectors"><span>Defining a Vector</span></a></li>
<li><a class="reference internal" href="#accessingvectorentries"><span>Accessing elements within a vector</span></a></li>
<li><a class="reference internal" href="#basicvectorops"><span>Basic operations on vectors</span></a></li>
</ol>
<div class="section" id="defining-a-vector">
<span id="definingvectors"></span><span id="index-0"></span><h2>Defining a Vector<a class="headerlink" href="#defining-a-vector" title="Permalink to this headline">¶</a></h2>
<p>Matlab is a software package that makes it easier for you to enter
matrices and vectors, and manipulate them. The interface follows a
language that is designed to look a lot like the notation use in
linear algebra. In the following tutorial, we will discuss some of the
basics of working with vectors.</p>
<p>If you are running windows or Mac OSX, you can start matlab by
choosing it from the menu. To start matlab on a unix system, open up a
unix shell and type the command to start the software: <em>matlab</em>. This
will start up the software, and it will wait for you to enter your
commands. In the text that follows, any line that starts with two
greater than signs (&gt;&gt;) is used to denote the matlab command line.
This is where you enter your commands.</p>
<p>Almost all of Matlab&#8217;s basic commands revolve around the use of
vectors. A vector is defined by placing a sequence of numbers within
square braces:</p>
<div class="highlight-matlab" id="index-1"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span><span class="p">]</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">3</span>     <span class="mi">1</span>
</pre></div>
</div>
<p>This creates a row vector which has the label &#8220;v&#8221;. The first entry in
the vector is a 3 and the second entry is a 1. Note that matlab
printed out a copy of the vector after you hit the enter key. If you
do not want to print out the result put a semi-colon at the end of the
line:</p>
<span class="target" id="index-2"></span><div class="highlight-matlab" id="index-3"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span><span class="p">];</span>
<span class="o">&gt;&gt;</span>
</pre></div>
</div>
<p>If you want to view the vector just type its label:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">3</span>     <span class="mi">1</span>
</pre></div>
</div>
<p>You can define a vector of any size in this manner:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">7</span> <span class="o">-</span><span class="mi">21</span> <span class="mi">5</span> <span class="mi">6</span><span class="p">]</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">3</span>     <span class="mi">1</span>    <span class="mi">7</span>    <span class="o">-</span><span class="mi">21</span>    <span class="mi">5</span>    <span class="mi">6</span>
</pre></div>
</div>
<p>Notice, though, that this always creates a row vector. If you want to
create a column vector you need to take the transpose of a row vector.
The transpose is defined using an apostrophe (<em>&#8216;</em>):</p>
<span class="target" id="index-4"></span><div class="highlight-matlab" id="index-5"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">1</span> <span class="mi">7</span> <span class="o">-</span><span class="mi">21</span> <span class="mi">5</span> <span class="mi">6</span><span class="p">]</span><span class="o">&#39;</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">3</span>
     <span class="mi">1</span>
     <span class="mi">7</span>
   <span class="o">-</span><span class="mi">21</span>
     <span class="mi">5</span>
     <span class="mi">6</span>
</pre></div>
</div>
<p>A common task is to create a large vector with numbers that fit a
repetitive pattern. Matlab can define a set of numbers with a common
increment using colons. For example, to define a vector whose first
entry is <em>1</em>, the second entry is <em>2</em>, the third is <em>3</em>, and
sequentially through <em>8</em>, you enter the following:</p>
<span class="target" id="index-6"></span><div class="highlight-matlab" id="index-7"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">8</span><span class="p">]</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">1</span>     <span class="mi">2</span>     <span class="mi">3</span>     <span class="mi">4</span>     <span class="mi">5</span>     <span class="mi">6</span>     <span class="mi">7</span>     <span class="mi">8</span>
</pre></div>
</div>
<p>If you wish to use an increment other than one that you have to define
the start number, the value of the increment, and the last number. For
example, to define a vector that starts with <em>2</em> and ends in <em>4</em> with
steps of <em>0.25</em> you enter the following:</p>
<span class="target" id="index-8"></span><div class="highlight-matlab" id="index-9"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">:.</span><span class="mi">25</span><span class="p">:</span><span class="mi">4</span><span class="p">]</span>

<span class="n">v</span> <span class="p">=</span>

  <span class="n">Columns</span> <span class="mi">1</span> <span class="n">through</span> <span class="mi">7</span>

    <span class="mf">2.0000</span>    <span class="mf">2.2500</span>    <span class="mf">2.5000</span>    <span class="mf">2.7500</span>    <span class="mf">3.0000</span>    <span class="mf">3.2500</span>    <span class="mf">3.5000</span>

  <span class="n">Columns</span> <span class="mi">8</span> <span class="n">through</span> <span class="mi">9</span>

    <span class="mf">3.7500</span>    <span class="mf">4.0000</span>
</pre></div>
</div>
</div>
<div class="section" id="accessing-elements-within-a-vector">
<span id="accessingvectorentries"></span><h2>Accessing elements within a vector<a class="headerlink" href="#accessing-elements-within-a-vector" title="Permalink to this headline">¶</a></h2>
<p>You can view individual entries in this vector. For example to view
the first entry just type in the following:</p>
<span class="target" id="index-10"></span><span class="target" id="index-11"></span><div class="highlight-matlab" id="index-12"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">2</span>
</pre></div>
</div>
<p>This command prints out entry <em>1</em> in the vector. Also notice that a
new variable called <em>ans</em> has been created. Any time you perform an
action that does not include an assignment matlab will put the label
<em>ans</em> on the result.</p>
<p>To simplify the creation of large vectors, you can define a vector by
specifying the first entry, an increment, and the last entry. Matlab
will automatically figure out how many entries you need and their
values. For example, to create a vector whose entries are <em>0</em>, <em>2</em>,
<em>4</em>, <em>6</em>, and <em>8</em>, you can type in the following line:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">8</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">2</span>     <span class="mi">4</span>     <span class="mi">6</span>     <span class="mi">8</span>
</pre></div>
</div>
<p>Matlab also keeps track of the last result. In the previous example, a
variable <em>ans</em> is created. To look at the transpose of the previous
result, enter the following:</p>
<div class="highlight-matlab" id="index-13"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="nb">ans</span><span class="o">&#39;</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>
     <span class="mi">2</span>
     <span class="mi">4</span>
     <span class="mi">6</span>
     <span class="mi">8</span>
</pre></div>
</div>
<p>To be able to keep track of the vectors you create, you can give them
names. For example, a row vector <strong>v</strong> can be created:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">8</span><span class="p">]</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">2</span>     <span class="mi">4</span>     <span class="mi">6</span>     <span class="mi">8</span>

<span class="o">&gt;&gt;</span> <span class="n">v</span>

<span class="n">v</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">2</span>     <span class="mi">4</span>     <span class="mi">6</span>     <span class="mi">8</span>

<span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">;</span>
<span class="o">&gt;&gt;</span> <span class="n">v</span><span class="o">&#39;</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>
     <span class="mi">2</span>
     <span class="mi">4</span>
     <span class="mi">6</span>
     <span class="mi">8</span>
</pre></div>
</div>
<p>Note that in the previous example, if you end the line with a semi-
colon, the result is not displayed. This will come in handy later when
you want to use Matlab to work with very large systems of equations.</p>
<p>Matlab will allow you to look at specific parts of the vector. If you
want to only look at the first three entries in a vector you can use
the same notation you used to create the vector:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">)</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">2</span>     <span class="mi">4</span>

<span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">)</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">4</span>

<span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">)</span><span class="o">&#39;</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>
     <span class="mi">4</span>
</pre></div>
</div>
</div>
<div class="section" id="basic-operations-on-vectors">
<span id="basicvectorops"></span><h2>Basic operations on vectors<a class="headerlink" href="#basic-operations-on-vectors" title="Permalink to this headline">¶</a></h2>
<p>Once you master the notation you are free to perform other operations:</p>
<span class="target" id="index-14"></span><div class="highlight-matlab" id="index-15"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">v</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">v</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">)</span>

<span class="nb">ans</span> <span class="p">=</span>

    <span class="o">-</span><span class="mi">2</span>    <span class="o">-</span><span class="mi">2</span>    <span class="o">-</span><span class="mi">2</span>
</pre></div>
</div>
<p>For the most part Matlab follows the standard notation used in linear
algebra. We will see later that there are some extensions to make some
operations easier. For now, though, both addition subtraction are
defined in the standard way. For example, to define a new vector with
the numbers from <em>0</em> to <em>-4</em> in steps of <em>-1</em> we do the following:</p>
<div class="highlight-matlab" id="index-16"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">u</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="mi">4</span><span class="p">]</span>
<span class="n">u</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span>

<span class="n">u</span> <span class="p">=</span>

     <span class="mi">0</span>    <span class="o">-</span><span class="mi">1</span>    <span class="o">-</span><span class="mi">2</span>    <span class="o">-</span><span class="mi">3</span>    <span class="o">-</span><span class="mi">4</span>
</pre></div>
</div>
<p>We can now add <strong>u</strong> and <strong>v</strong> together in the standard way:</p>
<div class="highlight-matlab" id="index-17"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">u</span><span class="o">+</span><span class="n">v</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">1</span>     <span class="mi">2</span>     <span class="mi">3</span>     <span class="mi">4</span>
</pre></div>
</div>
<p>Additionally, scalar multiplication is defined in the standard way.
Also note that scalar division is defined in a way that is consistent
with scalar multiplication:</p>
<div class="highlight-matlab" id="index-18"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">u</span>

<span class="nb">ans</span> <span class="p">=</span>

     <span class="mi">0</span>     <span class="mi">2</span>     <span class="mi">4</span>     <span class="mi">6</span>     <span class="mi">8</span>

<span class="o">&gt;&gt;</span> <span class="n">v</span><span class="o">/</span><span class="mi">3</span>

<span class="nb">ans</span> <span class="p">=</span>

         <span class="mi">0</span>    <span class="mf">0.6667</span>    <span class="mf">1.3333</span>    <span class="mf">2.0000</span>    <span class="mf">2.6667</span>
</pre></div>
</div>
<p>With these definitions linear combinations of vectors can be easily
defined and the basic operations combined:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">u</span><span class="o">+</span><span class="n">v</span><span class="o">/</span><span class="mi">3</span>

<span class="nb">ans</span> <span class="p">=</span>

         <span class="mi">0</span>    <span class="mf">2.6667</span>    <span class="mf">5.3333</span>    <span class="mf">8.0000</span>   <span class="mf">10.6667</span>
</pre></div>
</div>
<p>You will need to be careful. These operations can only be carried out
when the dimensions of the vectors allow it. You will likely get used
to seeing the following error message which follows from adding two
vectors whose dimensions are different:</p>
<div class="highlight-matlab"><div class="highlight"><pre><span class="o">&gt;&gt;</span> <span class="n">u</span><span class="o">+</span><span class="n">v</span><span class="o">&#39;</span>
??? <span class="n">Error</span> <span class="n">using</span> <span class="o">==&gt;</span> <span class="n">plus</span>
<span class="n">Matrix</span> <span class="n">dimensions</span> <span class="n">must</span> <span class="n">agree</span><span class="p">.</span>
</pre></div>
</div>
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